Independent Variable

Independent variable is complementary to dependent variable. These two concepts are used primarily in their mathematical sense, meaning that the value of a dependent variable changes in response to that of an independent variable. In research design, independent variables are those that a researcher can manipulate, whereas dependent variables are the responses to the effects of independent variables. By purposefully manipulating [Page 592]the value of an independent variable, one hopes to cause a response in the dependent variable.

As such, independent variables might carry different names in various research fields, depending on how the relationship between the independent and the dependent variable is defined. They might be called explanatory variables, controlled variables, input variables, predictor variables, factors, treatments, conditions, or other names. For instance, in regression experiments, they often are called regressors in relation to the regressand, the dependent, or the response variable.

The concept of independent variable in statistics should not be confused with the concept of independent random variable in probability theories. In the latter case, two random variables are said to be independent if and only if their joint probability is the product of their marginal probabilities for every pair of real numbers taken by the two random variables. In other words, if two random variables are truly independent, the events of one random variable have no relationship with the events of the other random variable. For instance, if a fair coin is flipped twice, a head occurring in the first flip has no association with whether the second flip is a head or a tail because the two events are independent.

Mathematically, the relationship between independent and dependent variables might be understood in this way:

where x is the independent variable (i.e., any argument to a function) and y is the dependent variable (i.e., the value that the function is evaluated to). Given an input of x, there is a corresponding output of y,x changes independently, whereas y responds to any change in x.

Equation 1 is a deterministic model. For each input in x, there is one and only one response in y. A familiar graphic example is a straight line if there is only one independent variable of order 1 in the previous model. In statistics, however, this model is grossly inadequate. For each value of x, there is often a population of y, which follows a probability distribution. To reflect more accurately this reality, the preceding equation is revised accordingly:

where E(y) is the expectation of y, or equivalently,

where ε is a random variable, which follows a specific probability distribution with a zero mean. This is a probabilistic model. It is composed of a deterministic part [f/(x)] and a random part (ε). The random part is the one that accounts for the variation in y.

In experiments, independent variables are the design variables that are predetermined by researchers before an experiment is started. They are carefully controlled in controlled experiments or selected in observational studies (i.e., they are manipulated by the researcher according to the purpose of a study). The dependent variable is the effect to be observed and is the primary interest of the study. The value of the dependent variable varies subjecting to the variation in the independent variables and cannot be manipulated to establish an artificial relationship between the independent and dependent variables. Manipulation of the dependent variable invalidates the entire study.

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